Pricing Rainbow Options Peter Ouwehand, Department of Mathematics and Applied Mathematics, University of Cape Town, South Africa E-mail address: [email protected] Graeme West, School of Computational & Applied Mathematics, University of the Witwatersrand, South Africa, Financial Modelling Agency, South Africa. www.finmod.co.za E-mail address: [email protected] Abstract Keywords A previous paper (West 2005) tackled the issue of calculating accurate uni-, bi- and exotic option, Black-Scholes model, exchange option, rainbow option, equivalent trivariate normal probabilities. This has important applications in the pricing of multi- martingale measure, change of numeraire, trivariate normal. asset options, e.g. rainbow options. In this paper, we derive the Black—Scholes prices of several styles of (multi-asset) rainbow options using change-of-numeraire machinery. Hedging issues and deviations from the Black-Scholes pricing model are also briefly considered. 1. Definition of a Rainbow Option • “Put 2 and call 1”, an exchange option to put a predefined risky asset and call the other risky asset, (Margrabe 1978). Thus, asset 1 is Rainbow Options refer to all options whose payoff depends on more called with the ‘strike’ being asset 2. than one underlying risky asset; each asset is referred to as a colour of Thus, the payoffs at expiry for rainbow European options are: the rainbow. Examples of these include: Best of assets or cash max(S , S ,...,S , K) • “Best of assets or cash” option, delivering the maximum of two risky 1 2 n Call on max max(max(S , S ,...,S ) − K, 0) assets and cash at expiry (Stulz 1982), (Johnson 1987), (Rubinstein 1 2 n Call on min max(min(S , S ,...,S ) − K, 0) 1991) 1 2 n Put on max max(K − max(S , S ,...,S ), 0) • “Call on max” option, giving the holder the right to purchase the max- 1 2 n Put on min max(K − min(S , S ,...,S ), 0) imum asset at the strike price at expriry, (Stulz 1982), (Johnson 1987) 1 2 n Put 2 and Call 1 max(S − S , 0) • “Call on min” option, giving the holder the right to purchase the 1 2 minimum asset at the strike price at expiry (Stulz 1982), (Johnson To be true to history, we deal with the last case first. 1987) • “Put on max” option, giving the holder the right to sell the maxi- 2. Notation and Setting mum of the risky assets at the strike price at expiry, (Margrabe 1978), (Stulz 1982), (Johnson 1987) Define the following variables: • “Put on min” option, giving the holder the right to sell the minimum • Si = Spot price of asset i, of the risky assets at the strike at expiry (Stulz 1982), (Johnson 1987) • K = Strike price of the rainbow option, 2 Wilmott magazine TECHNICAL ARTICLE 1 • σi = volatilityof asset i, currency. The risk free rate in this market is q2. Thus we have the option • qi = dividend yield of asset i, to buy asset one for a strike of 1. This has a Black-Scholes price of • ρij = correlation coefficient of return on assets i and j, = S1 −q τ −q τ • r the risk-free rate (NACC), V = e 1 N(d+ ) − e 2 N(d− ) • τ = the term to expiry of the rainbow option. S2 S1 Our system for the asset dynamics will be S 1 ln 2 + q − q ± σ 2 τ 1 2 1 2 d± = √ dS/S = (r − q)dt + AdW (1) σ τ where σ is the volatility of S1 . To get from a price in the new asset 2 cur- where the Brownian motions are independent. A is a square root of the S2 covariance matrix , that is AA = . As such, A is not uniquely deter- rency to a price in the original economy, we multiply by S2: the ‘exchange mined, but it would be typical to take A to be the Choleski decomposition rate’, which gives us (2). matrix of (that is, A is lower triangular). Under such a condition, A is So, what is σ ? We show that (5) is the correct answer to this question uniquely determined. in §4. th Let the i row of A be ai. We will say that ai is the volatility vector for asset Si. Note that if we were to write things where Si had a single volatil- 4. Change of Numeraire 2 n 2 ity σi then σ = = a , so σ i =a , where the norm is the usual i j 1 ij i Suppose that X is a European—style derivative with expiry date T. Since Euclidean norm. Also, the correlation between the returns of Si and Sj is a ·a i j (Harrison & Pliska 1981) it has been known that if X can be perfectly given by a a . i j hedged (i.e. if there is a self—financing portfolio of underlying instruments which perfectly replicates the payoff of the derivative at expiry), then the time—t value of the derivative is given by the following risk—neutral valua- 3. The Result of Margrabe tion formula: The theory of rainbow options starts with (Margrabe 1978) and has its = −r(T−t)EQ most significant other development in (Stulz 1982). Xt e t [XT ] (Margrabe 1978) began by evaluating the option to exchange one asset Q where r is the riskless rate, and the symbol E denotes the expectation at for the other at expiry. This is justifiably one of the most famous early op- t time t under a risk—neutral measure Q. A measure Q is said to be risk—neutral tion pricing papers. This is conceptually like a call on the asset we are ¯ −rt if all discounted asset prices St = e St are martingales under the measure going to receive, but where the strike is itself stochastic, and is in fact the ¯ Q, i.e. if the expected value of each St at an earlier time u is its current second asset. The payoff at expiry for this European option is: ¯ value Su: Q max(S1 − S2, 0), E ¯ = ¯ ≤ ≤ u [St] Su whenever 0 u t which can be valued as: (Here we assume for the moment that S pays no dividends.) rt Now let At = e denote the bank account. Then the above can be − − = q1 τ − q2 τ VM S1e N(d+ ) S2e N(d− ), (2) rewritten as X X where t = EQ T ¯ = EQ ¯ t i.e. Xt t XT f 1 At AT ln 1 ± σ 2τ f2 2 ¯ d± = √ (3) Thus X is a Q–martingale. σ τ t In an important paper, (Geman, El Karoui & Rochet 1995) it was shown that there is “nothing special” about the bank account: given an Aˆ Aˆ (r−qi )τ asset (1995) , we can “discount” each underlying asset using : fi = Sie (4) ˆ St St = Aˆ 2 = 2 + 2 − t σ σ1 σ2 2ρσ1σ2 (5) Thus Sˆ is the “price” of S measured not in money, but in units of Aˆ . Margrabe derives this formula by developing and then solving a The asset Aˆ is referred to as a numéraire, and might be a portfolio or a de- ˆ Black-Scholes type differential equation. But he also gives another argu- rivative—the only restriction is that its value At is strictly positive during ment, which he credits to Stephen Ross, which with the hindsight of the time period under consideration. modern technology, would be considered to be the most appropriate ap- It can be shown (cf. (Geman, et al. 1995)) that in the absence of arbi- proach to the problem. Let asset 2 be the numeraire in the market. In trage, and modulo some technical conditions, there is for each numéraire ^ other words, asset 2 forms a new currency, and asset one costs S1 in that (1995) Aˆ a measure Qˆ with the property that each numéraire—deflated S2 Wilmott magazine 3 ˆ ˆ Q ( ) = ( ) asset price process St is a Q—martingale, i.e. However, when we change to measure j, we know that Y t Si/j t (qi −q j )t e is a Q j—martingale. Applying Ito’s formula again, we see that the ˆ EQ ˆ = ˆ ≤ ≤ u [St] Su whenever 0 u t risk—neutral dynamics of Yt are given by ˆ dY (Again, we assume that S pays no dividends.) We call Q the equivalent = 2 − · + − · a j a i a j dt (a i a j) dW martingale measure (EMM) associated with the numéraire Aˆ. It then follows Y easily that if a European—style derivative X can be perfectly hedged, then Q Q Since Y(t) is a j—martingale, its drift under j is zero, and its volatil- Qˆ Qˆ X Q Y(t) ˆ = E ˆ = ˆ E T ity remains unchanged. Thus the j—dynamics of are Xt t [XT ] and so Xt At t ˆ AT dY = − · j = (ai aj) dW Indeed, if Vt is the value of a replicating portfolio, then (1) Xt Vt by Y ˆ = Vt Qˆ the law of one price, and (2) Vt ˆ is a —martingale. Thus At j where W is a standard n–dimensional Qj—Brownian motion. Applying −(q −q )t ˆ ˆ Ito’s formula once again to S (t) = Y(t)e i j , it follows easily that the ˆ = ˆ = EQ ˆ = EQ ˆ i/j Xt Vt t VT t XT Qj—dynamics of Si/j(t) are given by = dS using the fact that VT XT —by definition of “replicating portfolio”. i/j = − + − · j (qj qi) dt (a i a j) dW It follows that if N1, N2 are numéraires, with associated EMM’s Q1 , Q2 , Si/j then 2 = − 2 = 2 + 2 − Returning to §3, we have σ a 1 a 2 a 1 a 2 2ρ a 1 Q XT Q XT 2 2 E 1 = E 2 a =σ + σ − 2ρσ1σ2 , as required.